Bridging the Gap between Reactivity, Contraction, and Finite-Time Lyapunov Exponents

TL;DR

This paper reviews and extends the concepts of reactivity, contractivity, and finite-time Lyapunov exponents in discrete-time dynamical systems, establishing their interconnections and applications to network synchronization stability.

Contribution

It introduces a unified framework linking reactivity, contractivity, and Lyapunov exponents, and applies it to analyze stability and synchronization in complex networks.

Findings

Contractive p-iteration systems imply stable attractors.

Connections established among reactivity, contractivity, and Lyapunov exponents.

Framework applicable to time-varying and nonlinear maps.

Abstract

Reactivity, contractivity, and Lyapunov exponents are powerful tools for studying the stability properties of dynamical systems and have been extensively investigated in the literature for decades. In this paper, we review and extend the concepts of reactivity, contractivity, and finite-time Lyapunov exponents for discrete-time dynamical systems and establish connections among them. We focus on time-invariant maps, time-varying linear maps, and certain classes of time-varying nonlinear maps. In particular, we show that if the corresponding $p$-iteration systems (with p > 1) are contractive, then the original systems admit stable attractors such as fixed points or limit cycles. We demonstrate the application of these results to the analysis of synchronization stability in coupled networks and discuss how p-iteration systems can serve as a useful framework for studying network synchronization.